An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity

A novel constitutive formulation is developed for finitely deforming hyperelastic materials that exhibit isotropic behavior with respect to a reference configuration. The strain energy per unit reference volume, W, is defined in terms of three natural strain invariants, K1-3, which respectively specify the amount-of-dilatation, the magnitude-of-distortion, and the mode-of-distortion. Distortion is that part of the deformation that does not dilate. Moreover, pure dilatation (K-2=0), pure shear (K-3=0), uniaxial extension (K-3=1), and uniaxial contraction (K-3=-1) are tests which hold a strain invariant constant. Through an analysis of previously published data, it is shown for rubber that this new approach allows W to be easily determined with improved accuracy. Albeit useful for large and small strains, distinct advantage is shown for moderate strains (e.g. 2-25%). Central to this work is the orthogonal nature of the invariant basis. If eta represents natural strain, then {K-1,K-2,K-3} are such that the tensorial contraction of (partial derivativeK(i)/partial derivative (eta)) with (partial derivativeK(j)/partial derivative (eta)) vanishes when i not equalj. This result, in turn, allows the Cauchy stress t to be expressed as the sum of three response terms that are mutually orthogonal. In particular (summation implied) t=A(i)partial derivativeW/partial derivativeK(i), where the partial derivativeW/partial derivativeK(i) are scalar response functions and the A(i) are kinematic tensors that an mutually orthogonal, (C) 2000 Elsevier Science Ltd. All rights reserved.